Institute of Actuaries of India

Transcription

1 Insttute of ctuares of Inda Subject CT8-Fnancal Economcs ay 008 Examnaton INDICTIVE SOLUTION

2 II CT Q.1 a F0,5,6 1/6-5*ln0,5/0,6 Where, F0,5,6 s forard rate at tme 0 for delvery beteen tme 5 and 6 0,5 s zero coupon bond prce at tme 0 for e 1 payable at tme 5 0,6 s zero coupon bond prce at tme 0 for e 1 payable at tme 6 Therefore, F0,5,6 1/1* ln0.7/ % b arket prce of rsk: arket prce of the rsk represents the excess expected return over the rskfree rate per unt of volatlty n return for an nvestor takng on ths volatlty. γt,t mt,t rt/st,t c One-factor model: one-factor s one n hch nterest rates are assumed to be nfluenced by a sngle source of randomness The prces of all bonds of all maturtes and nterest rate dervatves must therefore move together. The randomness s usually modeled as an Itό process. The stochastc dfferental equaton for rt has the follong form under the real-orld probablty measure P: drt at,rtdt bt,rtdwt Where a. and b. are approprately chosen functons Lmtatons: The prces of bonds of dfferent terms n the real orld are not observed to be perfectly correlated, alays movng together. Sustaned perods have been observed hstorcally th all combnatons of hgh/lo nterest rates and hgh/lo volatlty. Ths seems to be nconsstent th one-factor model. Some nterest rate products are explctly dependent on other varables, hch ould be expected to ntroduce a separate source of randomness. Total arks: 9 Page of 15

3 II CT Q. a y put-call party e kno that c t k * exp r * Tt p t s t therefore, p t ,50 * exp , b Impled volatlty: c t St * Φd 1 K*expr * Tt * Φd here d 1 {logst/k r ½ * T- t} / * T-t d d 1 * T-t f 0.15, c t as d and so Φd and d and so Φd f 0.18, c t 00.7 as d and so Φd and d and so Φd k 5,50 s t 5,000 r 0.05 k * expr * Tt y lnear nterpolaton, 0.17 at 0.17, c t , d and so Φd and d and so Φd can therefore prce a call th strke of 4,750 and volatlty 0.17 usng formula n b Page 3 of 15

4 II CT c t k 4, as d and so Φd and d and so Φd p t k 4, k 4,750 s t 5,000 r 0.05 k * expr * Tt Total arks: 10 Q.3 1a One-step transton probablty matrx s: P State1 State State3 State4 State State State State P Page 4 of 15

5 II CT P Usng the probabltes calculated above, the rsk-neutral expected amounts of the payments are: Tme 1: 6.5* *0.75*0. 6.5*0.5* Tme : 6.5* *0.75* *0.5* Tme : 116.5* *0.75* *0.5* Present values of these rsk-neutral expected amounts: 5.55/1.065^ /1.065^ 78.70/1.065^ b The rsk-neutral present value calculated above s the far value of the bonds. Therefore, f the fnancal nsttuton buys the bonds at par s. 100, t ll be payng too much of the prce. The rate of return the fnancal nsttuton ll earn f the bond ssuer makes payments n full. 6.5/1^1 6.5/1^ 116.5/1^ Page 5 of 15

6 II CT y tral and mprovement, 11.5% The credt spread s the dfference beteen ths and the yeld on correspondng default-free bond. So, credt spread % Q.4 recombnng bnomnal tree or bnomnal lattce s one n hch the szes of the up-steps and don-steps are assumed to be the same under all states and across all tme ntervals..e., u t ju and d t jd for all tmes t and states j, th d < expr < u Total arks: 10 It therefore follos that the rsk neutral probablty q s also constant at all tmes and n all states eg. q t jq The man advantage of a n perod recombnng bnomnal tree s that t has only [n1] possble states of tme as opposed to n possble states n a smlar non-recombnng bnomnal tree. Ths greatly reduces the amount of computaton tme requred hen usng a bnomnal tree model. The man ds-advantage s that the recombnng bnomnal tree mplctly assumes that the volatlty and drft parameters of the underlyng asset prce are constant over tme, hch assumpton s contradcted by emprcal evdence. Page 6 of 15

7 II CT b The rsk-neutral probabltes at the frst and second steps are as follos: q 1 exp / / q exp / Put payoffs at the expraton date at each of the four possble states of expry are 0,0,0 and 95. Workng backards, the value of the opton V1 1 follong an up step over the frst 3 months s V1 1 exp0.05 [0.4177* 0] [0.588*0].e., V1 1 0 The value of the opton V1 follong a don step over the frst 3 months s: V1 exp0.05 [0.4177* 0] [0.588*95].e., V Page 7 of 15

8 II CT The current value of the put opton s: V0 exp [0.4510*0] [0.5490* ].e., V b. Whle the proposed modfcaton ould produce a more accurate valuaton, there ould be a lot more parameter values to specfy. pproprate values of u and d ould be requred for each branch of the tree and values of r for each month ould be requred. The ne tree ould have 6 64 nodes n the expry column. Ths ould render the calculatons prohbtve to do normally, and ould requre more programmng and calculaton tme on the computer. n alternatve model that mght be more effcent numercally ould be a 6-step recombnng tree hch ould have only 7 nodes n the fnal column. Total arks: 14 Page 8 of 15

9 II CT Q.5 a n arbtrage opportunty s a stuaton here sure proft can be made th no rsk. We can start at tme 0 th a portfolo that has a net value of zero. t some future tme T: The probablty of a loss s 0 The probablty to make strctly postve proft s greater than 0. In an effcent market t s dffcult to fnd an arbtrage opportunty because all the actve partcpants n the market ould aval ths opportunty and soon the market prces of the assets ould change to remove the arbtrage opportunty. b The 1 month cost of borrong money to buy 1 share: exp6/100* rbtrage opportunty: orro s for one month and buy 1 share of Infosys Sell 1 future contract of Infosys at s On 31 ay, 008, sell 1 share of Infosys and buy 1 future contract of Infosys Proft after I month ll be s..88 c Loer bound: Consder a portfolo, consstng of a European put opton on a non-dvdend payng share and a share. Compare ths th the alternatve of cash, currently orth K*exp-rT-t, here r : rsk-free nterest rate T-t : Tme to expry K : Strke prce St : Underlyng share prce at tme t t tme T, portfolo ll be orth at least as much as the cash, because, t tme T, cash ll be orth K. Page 9 of 15

10 II CT Portfolo the share plus the put opton ll be orth: K f S T < K because the opton ll be exercsed by sellng the share, leavng K S T f S T > K because the opton ll not be exercsed Thus the portfolo s alays orth at least as much as the cash depost at tme T. Therefore, p t St K*exp-rT-t So, p t K*exp-rT-t St d Upper bound: For an European put, the maxmum value obtanable at expry s the strke prce K. Therefore, the current value must satsfy: p t K*exp-rT-t It can t exceed the dscounted value of sum receved on exercse, hch t ll equal f the share prce falls to zero. Consder to portfolos: Portfolo : n European call opton plus cash orth K*exp-rT-t. The value of portfolo at the expry date ll be: K f S T K because the opton ll not be exercsed S T f S T > K because the opton ll be exercsed Portfolo : Underlyng share plus an European put opton th the same expry date and exercse prce as the call. The value of portfolo at the expry date ll be: K f S T K because the opton ll be exercsed S T f S T > K because the opton ll not be exercsed Page 10 of 15

11 II CT Thus, the values at expry are the same for both portfolos regardless of the share prce at that tme, e maxk, S T. Snce they have the same value at expry and snce the optons can t be exercsed before then they should have the same value at any tme t<t. Therefore, c t K*exp-rT-t p t St Ths relatonshp s knon as put-call party. Total arks: 16 Q 6 Soluton a Explan the dfferent forms of effcent market hypothess EH. The dfferent forms of EH are Strong Form arket prces ncorporate all nformaton avalable to the publc and nsders. Sem-strong form arket prces ncorporate all publcly avalable nformaton Weak form arket prces ncorporate all nformaton of hstorcal prces b Does eak form of EH mply that the strong form s applcable? Does the strong form of EH mply that the eak form s applcable? Explan. Hstorcal data s publcly avalable nformaton. Thus, the strong form mples that the eak form apples. Hoever, data other than hstorcal prces may mpact stock prces, e.g. plans for ntroducton of a ne product that may not be knon to the publc. Thus, eak form need not mply that the strong form holds. c Ho do the follong relate to the EH? - Techncal nalyss - Fundamental nalyss - Insder Tradng - Weak form of EH mples that the study of past prces cannot be used to predct future prce movements. Ths challenges the effectveness of techncal analyss. Smlarly, f techncal analyss orks, t mples that the market s not even eak form effcent. - Fundamental analyss reles on publcly avalable nformaton for predctng stocks that are underprced or overprced. The sem-strong form of EH thus challenges the effectveness of fundamental analyss. - Insder tradng regulatons mply that prce senstve nformaton exsts and s not Page 11 of 15

12 II CT reflected n the prces. Ths suggests that the strong form of EH may not hold. It can also be argued that snce nsder tradng s llegal, the nformaton cannot be acted upon and the market s effcent because of legslaton. 7a 100 P x 4000 ; here x s the unform 1,10 dstrbuton Varance of returns 10000varx varx Ex 4 [Ex ] Ex 110/ 11/ Varx 81/1 7/4 Ex varx [Ex] 7/4 11/ Ex 4 4 x dx / varx Varance of returns E P 100Ex Donsde sem-varance of returns x 4000 dx Donsde sem-varance of returns b The value of x for Px < Var at 10% confdence nterval Total arks: 9 8a Total arks: 11 Let the proporton of securty and securty n the portfolo be and respectvely. The expected return of the portfolo s gven by: E p E E The varance of returns of the portfolo s gven by Var Cov, p Snce the correlaton s -1, Cov, 1 Page 1 of 15

13 II CT Page 13 of 15 and, p Var For the portfolo to be rskless, p Var 0 or, 1 or, % 10% 10% Therefore, t s possble to construct a rskless portfolo. 8b The varance of the portfolo s gven by p Var 1 1 To fnd the proporton of securty n the portfolo that mnmzes the varance, e set 0 p Var δ δ. or, 0 4 or, or, Therefore, f 0.1, Proporton of securty n the portfolo for mnmum varance s 7.45% c The covarance beteen securtes I and j s gven by ], [ j j Cov C ], [ j j j Cov ε β α ε β α

14 II CT We kno that α, β, α j, β j are constants. Therefore, C j Cov β ε, β ε ] [ j j Cov β, β ] Cov[ β, ε ] Cov[ β, ε ] Cov[ ε, ε ] [ j j j j sngle ndex model assumes that Cov[ ε, ] 0 It also assumes that Cov ε, ε 0, hen j Therefore, C Cov[ β j β β Cov[ β β V j j j, β, j ] ] 9a Total arks: 1 Under CP, the rsk of a securty s measured as the varance of returns. The rsk comprses to components dversfable rsk and non-dversfable rsk. The dversfable rsk s the rsk specfc to the company or the ndustry and can be elmnated by sutable dversfcaton of the portfolo. The non-dversfable or the systematc rsk s component of the rsk because of the market as a hole and s measured as volatlty of returns of the securty relatve to the volatlty of the market as a hole. It s denoted by β. Under the CP, the market reards only non-dversfable rsk and not does not reard dversfable rsk. 9b Under CP, the market prce of rsk s defned as Where E expected return on the market portfolo rsk free rate of return standard devaton of returns from the market E r ; Page 14 of 15

15 II CT It ndcates the addtonal expected return that the market requres for acceptng an addtonal unt of rsk as measured by the standard devaton of the market returns. E [5,000. 1%.4 3%.4 6% 75,000. %.4 5%.4 8%] 100, % 5,000 5,000 5,000 1% 75,000 % [ 4.45% 100,000 3% 75,000 5% 4.45% ,000 6% 75,000 8% 4.45% 0.4] 100, % E arket prce of rsk r 4.45% 3% 4.9% 3.38% 0. *********************** Total arks: 9 Page 15 of 15